Finite Group Databasebeta

← C31×#11694 #21468453

Subgroups — C31×#11694 #21468453

Order 7936.

468 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2621C211yesmaximal, characteristic
2C31×#1740C211yesmaximal, characteristic
3C31×#1458C211yesmaximal, characteristic
4C31×#1344C214yesmaximal
5C2×C31×C4⋊Q8C411yescharacteristic, complemented
6C2×C31×C22⋊Q8—21no—
7C2×C31×#117—21no—
8C2×C31×#117C2211yescharacteristic
9C31×#431—21no—
10C31×#398—21no—
11C31×#398C412yescomplemented
12C31×#397—21no—
13C31×#397C2212yes—
14C31×#391—21no—
15C31×#382C411yescharacteristic, complemented
16C31×#342—21no—
17C31×#342C2212yes—
18C31×#324—21no—
19C31×#323C2212yes—
20C22×Q8×C31—21no—
21C23×C4×C31D411yescharacteristic
22C31×C4⋊Q8—41no—
23C31×C4⋊Q8—41no—
24C31×C22⋊Q8—42no—
25C31×C22⋊Q8—42no—
26C31×#117—24no—
27C31×#117—41no—
28C31×#117—41no—
29C31×#117—42no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—41no—
34C2×C4⋊C4×C31C2×C412yes—
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C4⋊C4×C31D411yescharacteristic
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31D411yescharacteristic
40C2×C22⋊C4×C31C2311yescharacteristic
41C2×C42×C31—41no—
42C2×C42×C31C2×C411yescharacteristic
43C2×C42×C31D411yescharacteristic
44C31×C4⋊C8—41no—
45C31×C22⋊C8—22no—
46C4×C8×C31—41no—
47C31×#95—21no—
48C31×#95—22no—
49C31×#95—22no—
50C31×#95C2×C412yes—
51C24×C31C2×D411yescharacteristic
52C2×Q8×C31—41no—
53C2×Q8×C31—41no—
54C2×Q8×C31—42no—
55C2×Q8×C31—42no—
56C22×C4×C31—21no—
57C22×C4×C31—21no—
58C22×C4×C31—22no—
59C22×C4×C31—22no—
60C22×C4×C31—22no—
61C22×C4×C31—42no—
62C22×C4×C31C22⋊C411yescharacteristic
63C22×C4×C31C22⋊C411yescharacteristic
64C22×C4×C31C22⋊C411yescharacteristic
65C22×C4×C31C22⋊C411yescharacteristic
66C22×C4×C31D812yes—
67C22×C4×C31SD1612yes—
68C22×C4×C31C22×C411yescharacteristic
69C22×C4×C31C2×D411yescharacteristic
70C2×C8×C31—42no—
71C4⋊C4×C31—41no—
72C4⋊C4×C31—41no—
73C4⋊C4×C31—42no—
74C4⋊C4×C31—44no—
75C4⋊C4×C31—42no—
76C4⋊C4×C31—42no—
77C4⋊C4×C31—44no—
78C22⋊C4×C31—24no—
79C22⋊C4×C31—41no—
80C22⋊C4×C31—41no—
81C22⋊C4×C31—42no—
82C22⋊C4×C31—42no—
83C22⋊C4×C31—42no—
84C42×C31—21no—
85C42×C31—21no—
86C42×C31—21no—
87C42×C31—21no—
88C42×C31—44no—
89#11694C3111yesmaximal, characteristic, direct product factor, Sylow
90C23×C31—21no—
91C23×C31—21no—
92C23×C31—41no—
93C23×C31—41no—
94C23×C31C2×C22⋊C411yescharacteristic
95C23×C31C2×D811yescharacteristic
96C23×C31C2×SD1611yescharacteristic
97Q8×C31—82no—
98Q8×C31—82no—
99C2×C4×C31—21no—
100C2×C4×C31—21no—
101C2×C4×C31—22no—
102C2×C4×C31—22no—
103C2×C4×C31—22no—
104C2×C4×C31—22no—
105C2×C4×C31—21no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—21no—
109C2×C4×C31—41no—
110C2×C4×C31—41no—
111C2×C4×C31—41no—
112C2×C4×C31—41no—
113C2×C4×C31—44no—
114C2×C4×C31—42no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—42no—
118C2×C4×C31—42no—
119C2×C4×C31—42no—
120C2×C4×C31—82no—
121C2×C4×C31—82no—
122C2×C4×C31#9911yescharacteristic
123C2×C4×C31#9911yescharacteristic
124C2×C4×C31#10212yes—
125C2×C4×C31#10212yes—
126C2×C4×C31#10412yes—
127C8×C31—82no—
128#2621C2×C3111yescharacteristic
129#1740C2×C3111yescharacteristic
130#1458C2×C3111yescharacteristic
131#1344C2×C3114yes—
132C22×C31—21no—
133C22×C31—21no—
134C22×C31—21no—
135C22×C31—21no—
136C22×C31—41no—
137C22×C31—41no—
138C22×C31—41no—
139C22×C31—41no—
140C22×C31—81no—
141C22×C31C2×#9911yescharacteristic
142C22×C31C2×#10211yescharacteristic
143C22×C31C2×#10411yescharacteristic
144C4×C31—22no—
145C4×C31—41no—
146C4×C31—41no—
147C4×C31—41no—
148C4×C31—41no—
149C4×C31—82no—
150C4×C31—84no—
151C4×C31—82no—
152C4×C31#32712yes—
153C4×C31#32712yes—
154C2×C4⋊Q8C4×C3111yescharacteristic, complemented
155C2×C22⋊Q8—21no—
156C2×#117—21no—
157C2×#117C22×C3111yescharacteristic
158#431—21no—
159#398—21no—
160#398C4×C3112yescomplemented
161#397—21no—
162#397C22×C3112yes—
163#391—21no—
164#382C4×C3111yescharacteristic, complemented
165#342—21no—
166#342C22×C3112yes—
167#324—21no—
168#323C22×C3112yes—
169C2×C31—21no—
170C2×C31—21no—
171C2×C31—41no—
172C2×C31—41no—
173C2×C31C2×#32711yescharacteristic
174C2×C31#212311yescharacteristic
175C2×C31#212911yescharacteristic
176C22×Q8—21no—
177C23×C4D4×C3111yescharacteristic
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182#117—24no—
183#117—41no—
184#117—41no—
185#117—42no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×C4×C3112yes—
191C2×C4⋊C4C2×C4×C3111yescharacteristic
192C2×C4⋊C4D4×C3111yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4D4×C3111yescharacteristic
196C2×C22⋊C4C23×C3111yescharacteristic
197C2×C42—41no—
198C2×C42C2×C4×C3111yescharacteristic
199C2×C42D4×C3111yescharacteristic
200C4⋊C8—41no—
201C22⋊C8—22no—
202C4×C8—41no—
203#95—21no—
204#95—22no—
205#95—22no—
206#95C2×C4×C3112yes—
207C31#1169411yescharacteristic, direct product factor, Sylow
208C24C2×D4×C3111yescharacteristic
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4C22⋊C4×C3111yescharacteristic
220C22×C4C22⋊C4×C3111yescharacteristic
221C22×C4C22⋊C4×C3111yescharacteristic
222C22×C4C22⋊C4×C3111yescharacteristic
223C22×C4D8×C3112yes—
224C22×C4SD16×C3112yes—
225C22×C4C22×C4×C3111yescharacteristic
226C22×C4C2×D4×C3111yescharacteristic
227C2×C8—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C22⋊C4—24no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—44no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23C2×C22⋊C4×C3111yescharacteristic
251C23C2×D8×C3111yescharacteristic
252C23C2×SD16×C3111yescharacteristic
253Q8—82no—
254Q8—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4C31×#9911yescharacteristic
279C2×C4C31×#9911yescharacteristic
280C2×C4C31×#10212yes—
281C2×C4C31×#10212yes—
282C2×C4C31×#10412yes—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22C2×C31×#9911yescharacteristic
294C22C2×C31×#10211yescharacteristic
295C22C2×C31×#10411yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—82no—
302C4—84no—
303C4—82no—
304C4C31×#32712yes—
305C4C31×#32712yes—
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×C31×#32711yescharacteristic
311C2C31×#212311yescharacteristic
312C2C31×#212911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. C2×C31×C4⋊Q8C4semidirect
11. C31×#398C4semidirect
15. C31×#382C4semidirect
89. #11694C31direct (N × H)
154. C2×C4⋊Q8C4×C31semidirect
160. #398C4×C31semidirect
164. #382C4×C31semidirect
207. C31#11694direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C31×#26212
C31×#17402
C31×#14582
C31×#13442
#1169431